Question: $ E = \left[\begin{array}{rr}5 & 0 \\ -2 & 5\end{array}\right]$ $ C = \left[\begin{array}{rrr}-1 & 2 & 3 \\ 0 & 5 & 1\end{array}\right]$ What is $ E C$ ?
Answer: Because $ E$ has dimensions $(2\times2)$ and $ C$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ E C = \left[\begin{array}{rr}{5} & {0} \\ {-2} & {5}\end{array}\right] \left[\begin{array}{rrr}{-1} & \color{#DF0030}{2} & \color{#9D38BD}{3} \\ {0} & \color{#DF0030}{5} & \color{#9D38BD}{1}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ E$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ E$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ E$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{5}\cdot{-1}+{0}\cdot{0} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ E$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{5}\cdot{-1}+{0}\cdot{0} & ? & ? \\ {-2}\cdot{-1}+{5}\cdot{0} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ E$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{5}\cdot{-1}+{0}\cdot{0} & {5}\cdot\color{#DF0030}{2}+{0}\cdot\color{#DF0030}{5} & ? \\ {-2}\cdot{-1}+{5}\cdot{0} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{5}\cdot{-1}+{0}\cdot{0} & {5}\cdot\color{#DF0030}{2}+{0}\cdot\color{#DF0030}{5} & {5}\cdot\color{#9D38BD}{3}+{0}\cdot\color{#9D38BD}{1} \\ {-2}\cdot{-1}+{5}\cdot{0} & {-2}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{5} & {-2}\cdot\color{#9D38BD}{3}+{5}\cdot\color{#9D38BD}{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-5 & 10 & 15 \\ 2 & 21 & -1\end{array}\right] $